Beware though as this lecture was aimed at teachers. I must admit to not finding it particularly easy as it is a long time since I had to work with this sort of maths. To all school students reading this, let this be a lesson to you. If you don’t use it you lose it.

The science of light and colour is fantastically important in an enormous number of areas such as observing and understanding the universe, diagnosis and treatment processes in medicine, efficient communications and signal processing in industries. The talk looked at the basic concepts of the topic and showed how both the biological world and technology are making the most of light’s astonishing properties.

A complex number is a number that can be expressed in the form a + bi (or a +bj), where a and b are real numbers and i or (j) is the imaginary unit, where i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone. As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them “fictitious” during his attempts to find solutions to cubic equations in the 16th century, but complex numbers are no more or less “fictitious” or “imaginary” than any other kind of number.

If you are have only gone as far as GCSE maths you will have been told that you can’t take the square root of a negative number. That’s because you had no numbers which were negative after you’d squared them (so you couldn’t “go backwards” by taking the square root). Every number was positive after you squared it. So you couldn’t very well square-root a negative and expect to come up with anything sensible.

Now, however, you can take the square root of a negative number, but it involves using a new number to do it. This new number was invented (discovered?) around the time of the Reformation (see above). At that time, nobody believed that any “real world” use would be found for this new number, other than easing the computations involved in solving certain equations, so the new number was viewed as being a pretend number invented for convenience sake.

(But then, when you think about it, aren’t all numbers inventions? It’s not like numbers grow on trees! They live in our heads. We made them all up! Why not invent a new one, as long as it works okay with what we already have?)

Anyway, this new number was called “i“, standing for “imaginary”, because “everybody knew” that i wasn’t “real”. (That’s why you couldn’t take the square root of a negative number before: you only had “real” numbers; that is, numbers without the “i” in them.) The imaginary is defined to be:

Then:

Now, you may think you can do this:

But this doesn’t make any sense! You already have two numbers that square to 1; namely –1 and +1. And i already squares to –1. So it’s not reasonable that i would also square to 1. This points out an important detail: When dealing with imaginaries, you gain something (the ability to deal with negatives inside square roots), but you also lose something (some of the flexibility and convenient rules you used to have when dealing with square roots). In particular, YOU MUST ALWAYS DO THE i-PART FIRST!

(Remember imaginary numbers can be denoted with an i or a j)

At Rooks Heath A level students learn about simple harmonic motion in year 13 but they don’t need to know the maths seen during the lecture.

Complex numbers are important because they allow phase analysis. Φ is a phase term.

The word “Forced” is used because an object is forced to oscillate with simple harmonic motion. x is the displacement of the oscillator; A is the amplitude of the oscillation; e is the symbol for an exponential; x with a dot over it is the first differential, which in this case is the velocity of the oscillator; x with two dots over it is the second differential, which in this case is the acceleration of the oscillator; ω is the angular velocity of the oscillator; k is the stiffness constant.

Mechanical impedance is a measure of how much a structure resists motion when subjected to a given force.

The graph above is something that A level students are familiar with (but in not so much detail). If an oscillator is forced to oscillate at its natural frequency you get the maximum amplitude. However if you damp the oscillation you decrease the amplitude.

At school level the refractive index (n) is simply the speed of light in a vacuum/air divided by the speed of light in a material although students will normally calculate it by dividing sine of the angle of incidence in air by the sine of the angle of refraction in the material. The diagram below left shows refraction of light at the interface between two media of different refractive indices, with n2>n1. Since the phase velocity is lower in the second medium (v2<v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.

Electrical polarisability is the relative tendency of a charge distribution, like the electron cloud of an atom or molecule, to be distorted from its normal shape by an external electric field.

The permittivity of a medium describes how much electric field (more correctly, flux) is ‘generated’ per unit charge in that medium. Symbol: e
Unit: faradmeter-1 (F m-1). The permittivity of a substance (ε) is a characteristic which describes how it affects any electric field set up in it. A high permittivity tends to reduce any electric field present. In electromagnetism, permeability (µ) is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetisation that a material obtains in response to an applied magnetic field. In both cases if there is a 0 subscript then both are referring to a vacuum/air.

The negative sign on the permittivity is largely related to the “phase” between the applied electric field (such as is present for light that is incident on material) and the electron distribution in the material that is produced by the incident light (E field).

To understand this as a concept, think of Hooke’s Law F=-kx; the negative sign in this means that the applied force is out of phase (opposite direction) with the displacement it produces.

Back to ε now; the relation between applied E-field and the electric displacement (D) it produces is

D = ε times E.

So when ε (the permittivity) is negative for a range of frequencies, this infers the phase relation between the E-field and the displacement it produces in the electrons of the material (within this range of frequencies).

The above picture on the far left shows the usual effect of refraction where the straw looks bent. This is because light travels more slowly in water than air. The picture on the near left shows what a negative refractive index would give you, a more weird bent straw.

Plasma is one of the four fundamental states of matter (the others being solid, liquid, and gas). Heating a gas may ionise its molecules or atoms (reducing or increasing the number of electrons in them), thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions. Ionisation can be induced by other means, such as strong electromagnetic field applied with a laser or microwave generator, and is accompanied by the dissociation of molecular bonds, if present. Plasma can also be created by the application of an electric field on a gas, where the underlying process is the Townsend avalanche.

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in a conductor, but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced toward the field and negative charges shift in the opposite direction. This creates an internal electric field which reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarised, but also reorient so that their symmetry axis aligns to the field.

A magneto-optic effect is any one of a number of phenomena in which an electromagnetic wave propagates through a medium that has been altered by the presence of a quasistatic magnetic field. In such a material, which is also called gyrotropic or gyromagnetic, left- and right-rotating elliptical polarisations can propagate at different speeds, leading to a number of important phenomena. When light is transmitted through a layer of magneto-optic material, the result is called the Faraday effect: the plane of polarisation can be rotated, forming a Faraday rotator. The results of reflection from a magneto-optic material are known as the magneto-optic Kerr effect (not to be confused with the nonlinear Kerr effect).

The above picture shows a split-ring resonator array (metamaterial) arranged to produce a negative index of refraction, constructed of copper split-ring resonators and wires mounted on interlocking sheets of fibreglass circuit board.
The total array consists of 3 by 20×20 unit cells with overall dimensions of 10×100×100 millimetres. The height of 10 millimetres measures a little more than six subdivision marks on the ruler, which is marked in inches.Credit:NASA Glenn Research Center

Metamaterial summary:

The introduction of metamaterials in the 1990’s opened up new possibilities in electromagnetics;

There has been a successful implementation of metamaterial technology within the microwave spectrum;

Photonic crystals are periodic optical nanostructures that affect the motion of photons in much the same way that semiconductors affect electrons. Photonic crystals occur in nature in the form of structural coloration and promise to be useful in different forms in a range of applications.

Iridescence (also known as goniochromism) is generally known as the property of certain surfaces that appear to change colour as the angle of view or the angle of illumination changes. Examples of iridescence include soap bubbles, butterfly wings and sea shells.